We consider a heterotic version of six-dimensional Kodaira-Spencer gravityderived from the heterotic superpotential. We compute the one-loop partition function andfnd it can be expressed as a product of holomorphic Ray-Singer torsions. We discuss itstopological properties and potential gauge and gravitational anomalies. We show theseanomalies can be cancelled using Green-Schwarz-like counter-terms. We also discuss thedependence on the background geometry, and in particular the choice of hermitian metricneeded for quantisation. Given suitable topological constraints, this dependence may againbe cancelled by the addition of purely background-dependent counter-terms. We also explainhow our methods provide the one-loop partition functions of a large class of more generalholomorphic feld theories in terms of holomorphic Ray-Singer torsions.